350 



BAROMETRICAL MEASUREMENT OF HEIGHTS. 



introducing into this expression the value in metres of a, the earth's mean radius t 

 making z = Log. l-^-\ 18336 and Log. f -g-) = (j^) , which can be done without 

 sensible error, the above formula takes the following form, sufficiently accurate for 

 practical purposes : 



x = Log. (4). 18336 metres X -{ (1 + 0.0028371 cos. 2. L) 



> + l 



the four factors of which can easily be developed in tables, as has been done by Mr. 

 Oltmanns. But though this savant chose to develop also the second factor, I found 

 it better not to do so, partly because the calculation of it is very easy, and also on 

 account of the great extent it would have been necessary to give to this table, in 

 order to avoid troublesome interpolations. 



Tn the calculation of h'. (-^jgg-Ji Mr. Oltmanns used the constant coefficient of the 

 absolute expansion of the mercurial column ; I took that of the relative expansion 

 of the mercury and of the brass scale. It is obvious, therefore, that if the scale 

 of the barometer employed was of wood, glass, iron, or of another substance, it 

 would be necessary to make use of as many different coefficients, and the Table II. 

 could not be used. Moreover, Oltmanns combined the last two factors of the gen- 

 eral formula in one single table with double entry. This table I have calculated, 

 extending it sufficiently to avoid a double interpolation ; but as it seemed to me 

 much too extensive, I substituted for it Tables III. and IV., which are more condensed, 

 without rendering any troublesome interpolation necessary. 



I carried the calculation of these tables beyond the limits at which Olknanns chose 

 to stop, in order that they may answer for the most extreme cases. 



At the head of each table will be found the factor of which it is the development ; 

 this makes any other explanation superfluous. 



All these tables give, at sight, the numbers wanted ; only when very great pre- 

 cision is desired, a slight interpolation, ai sight, and very easy to apply, may be re- 

 quired. My principal object was to relieve the computer of the troublesome and 

 annoying labor of interpolations. 



1 added to these four tables the small Table V., taken from the Annuaire du 

 Bureau des Longitudes of Paris. It will be seldom used. 



When calculating differences of level, in the same order, with the tables, and by 

 the complete formula of Laplace, the results thus obtained never differ by more than 

 one decimetre in the most extreme cases. The following example will illustrate this 

 statement. I take the observation made in a balloon, by Gay-Lussac, at Paris, as 

 an extreme case, which is very well adapted to manifest the errors of the tables, 

 if there were any, by comparing the results obtained by means of them with those of 

 the direct calculation according to the complete formula of Laplace, from which 

 they are derived. 



D 10 



